Ordinary integers can only represent integral values. "Floating-point numbers" can represent non-integral values. This is useful for engineering, science, statistics, graphics, and any time you need to represent numbers from the real world, which are rarely integral!

Floats store numbers in an odd way--they're really storing the number in scientific notation, like

x = + 3.785746 * 10

Note that:

- You only need one bit to represent the sign--plus or minus.
- The exponent's just an integer, so you can store it as an integer.
- The 3.785746 part can be stored as the integer 3785746 (at least as long as you can figure out where the decimal point goes!)

x = + 3.785746 * 10

It's common to "normalize" a number in scientific notation so that:

- There's exactly one digit to the left of the decimal point.
- And that digit ain't zero.

In binary, a "normalized" number *always* has a 1 at the left of the decimal point (if it ain't zero, it's gotta be one). So there's no reason to even store the 1; you just know it's there!

(Note that there are also "denormalized" numbers, like 0.0, that don't have a leading 1. This is how zero is represented--there's an implicit leading 1 only if the exponent field is nonzero, an implicit leading 0 if the exponent field is zero...)

A "float" (as defined by IEEE Standard 754) consists of three bitfields:

Sign |
Exponent |
Fraction (or
"Mantissa") |

1 bit-- 0 for positive 1 for negative |
8 unsigned bits-- 127 means 2 ^{0
} 137 means 2^{10
} |
23 bits-- a binary fraction. Don't forget the implicit leading 1! |

The hardware interprets a float as having the value:

value = (-1)

Note that the mantissa has an implicit leading binary 1 applied (unless the exponent field is zero, when it's an implicit leading 0; a "denormalized" number).

For example, the value "8" would be stored with sign bit 0, exponent 130 (==3+127), and mantissa 000... (without the leading 1), since:

8 = (-1)

You can actually dissect the parts of a float using a "union" and a bitfield like so:

/* IEEE floating-point number's bits: sign exponent mantissa */(Executable NetRun link)

struct float_bits {

unsigned int fraction:23; /**< Value is binary 1.fraction ("mantissa") */

unsigned int exp:8; /**< Value is 2^(exp-127) */

unsigned int sign:1; /**< 0 for positive, 1 for negative */

};

/* A union is a struct where all the fields *overlap* each other */

union float_dissector {

float f;

float_bits b;

};

float_dissector s;

s.f=8.0;

std::cout<<s.f<<"= sign "<<s.b.sign<<" exp "<<s.b.exp<<" fract "<<s.b.fraction<<"\n";

return 0;

There are several different sizes of floating-point types:

C Datatype |
Size |
Approx. Precision |
Approx. Range |
Exponent Bits |
Fraction Bits |
+-1 range |

float |
4 bytes (everywhere) |
1.0x10^{-7} |
10^{38} |
8 |
23 |
2^{24} |

double |
8 bytes (everywhere) |
2.0x10^{-15} |
10^{308} |
11 |
52 |
2^{53} |

long double |
12-16 bytes (if it exists) |
2.0x10^{-20} |
10^{4932} |
15 |
64 |
2^{65} |

Nowadays floats have roughly the same performance as integers: addition takes a little over a nanosecond (slightly slower than integer addition); multiplication takes a few nanoseconds; and division takes a dozen or more nanoseconds. That is, floats are now cheap, and you can consider using floats for all sorts of stuff--even when you don't care about fractions.

for (int i=1;i<1000000000;i*=10) {(executable NetRun link)

double mul01=i*0.1;

double div10=i/10.0;

double diff=mul01-div10;

std::cout<<"i="<<i<<" diff="<<diff<<"\n";

}

On my P4, this gives:

i=1 diff=5.54976e-18That is, there's a factor of 10^-15 difference between double-precision 0.1 and the true 1/10!

i=10 diff=5.55112e-17

i=100 diff=5.55112e-16

i=1000 diff=5.55112e-15

i=10000 diff=5.55112e-14

i=100000 diff=5.55112e-13

i=1000000 diff=5.54934e-12

i=10000000 diff=5.5536e-11

i=100000000 diff=5.54792e-10

Program complete. Return 0 (0x0)

1.2347654 * 10

But to three decimal places,

1.234 * 10

which is to say, adding a tiny value to a great big value might not change the great big value at all, because the tiny value gets lost when rounding off to 3 places. This "roundoff" has implications.

For example, adding one repeatedly will eventually stop doing anything:

float f=0.73;(executable NetRun link)

while (1) {

volatile float g=f+1;

if (g==f) {

printf("f+1 == f at f=%.3f, or 2^%.3f\n",

f,log(f)/log(2.0));

return 0;

}

else f=g;

}

Recall that for integers, adding one repeatedly will *never* give you the same value--eventually the integer will wrap around, but it won't just stop moving like floats!

For another example, floating-point arithmetic isn't "associative"--if you change the order of operations, you change the result (up to roundoff):

1.2355308 * 10

1.2355308 * 10

In other words, parenthesis don't matter if you're computing the exact result. But to three decimal places,

1.235 * 10

1.234 * 10

In the first line, the small values get added together, and together they're enough to move the big value. But separately, they splat like bugs against the windshield of the big value, and don't affect it at all!

double lil=1.0;(executable NetRun link)

double big=pow(2.0,64);

printf(" big+(lil+lil) -big = %.0f\n", big+(lil+lil) -big);

printf("(big+lil)+lil -big = %.0f\n",(big+lil)+lil -big);

float gnats=1.0;(executable NetRun link)

volatile float windshield=1<<24;

float orig=windshield;

for (int i=0;i<1000;i++)

windshield += gnats;

if (windshield==orig) std::cout<<"You puny bugs can't harm me!\n";

else std::cout<<"Gnats added "<<windshield-orig<<" to the windshield\n";

In fact, if you've got a bunch of small values to add to a big value, it's more roundoff-friendly to add all the small values together first, then add them all to the big value:

float gnats=1.0;(executable NetRun link)

volatile float windshield=1<<24;

float orig=windshield;

volatile float gnatcup=0.0;

for (int i=0;i<1000;i++)

gnatcup += gnats;

windshield+=gnatcup; /* add all gnats to the windshield at once */

if (windshield==orig) std::cout<<"You puny bugs can't harm me!\n";

else std::cout<<"Gnats added "<<windshield-orig<<" to the windshield\n";

Roundoff is very annoying, but it doesn't matter if you don't care about exact answers, like in simulation (where "exact" means the same as the real world, which you'll never get anyway) or games.

This would be fine, except software is way slower than hardware--about 25x slower on the NetRun machine!

float f=pow(2,-128); // denormal(executable NetRun link)

int foo(void) {

f*=1.00001; //<- you can do almost any operation here...

return 0;

}

As written, this takes 328ns per execution of foo, which is crazy slow.

If you initialize f to, say, 3.0 (or any non-denormal value), this takes like 13ns per execution, which is reasonable.

That is, floating-point code can take absurdly longer when computing denormals (infinities have the same problem). Denormals can have a huge impact on the performance of real code--I've written a paper on this.

The easiest way to fix denormals is to round them off to zero--one trick for doing this is to just add a big value ("big" compared to a denormal can be like 1.0e-10) and then subtract it off again!